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I'm familiar with the Weierstrass approximation theorem and some aspects of the Stone-Weierstrass theorem but I mainly only get it for closed intervals [a, b]. I am familiar with the proof that begins with showing f is continuous on $[0, 1]$ and going from there. I have a 3-d set which forms an ellipsoid and I'd like to show that any continuous function on that set can also be approximated by a polynomial. Is there a way to extend the proof of Weierstrass approximation theorem or is this way over my head.

For example, [this post] (Showing a continuous functions on a compact subset of $\mathbb{R}^3$ can be uniformly approximated by polynomials) has an example set but I can't really follow the answer. I only know introductory real analysis. I'm guessing there's a way to go from the $[0, 1]$ case to the unit cube case but I'm missing that leap.

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  • $\begingroup$ "Prove any continuous function can be approximated" could mean "Pick any continuous function and prove that it can be approximated", or it could mean "Prove that any continuous function, regardless of which one it is, can be uniformly approximated". I think the latter is what you meant. Just changing "any" to "every" would remove all ambiguity. $\endgroup$ – Michael Hardy May 4 '14 at 4:21
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The Stone-Weierstrass theorem says that any closed subalgebra of $C(S)$ (where $S$ is any compact Hausdorff space) that separates points, contains the constants and is closed under complex conjugation is all of $C(S)$. In this case your ellipsoid $S$ is a compact Hausdorff space. Let $A$ be the uniform closure in $C(S)$ of the polynomials in $x, y, z$. This satisfies all requirements of the Stone-Weierstrass theorem, so it is $C(S)$, i.e. every continuous function on $S$ can be uniformly approximated by polynomials.

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